The conventional $\textrm{e}^N$ laminar-to-turbulent transition-prediction method focuses on the relative growth rate, called the $N$ factor, and neglects receptivity. To improve predictions, Mack (1977) proposed the amplitude method to incorporate receptivity, nonlinear effects and broadband characteristics. Currently, the lack of accurate receptivity coefficients, estimates of initial disturbance amplitudes at the lower-branch neutral position, referred to as branch I (where the imaginary part of the spatial wavenumber is zero), hinders the application of the amplitude method. Although experimental- and numerical-receptivity analyses have been conducted previously, they rely on correlations or indirect approaches. For the purpose of direct evaluation, this study applies bi-orthogonal decomposition to direct numerical simulation (DNS) data of a hypersonic boundary layer over a blunt cone, extracting initial amplitudes of instability modes. The decomposition framework incorporates both boundary-layer and entropy-layer modes, enabling direct evaluation of receptivity coefficients at branch I. The decomposed modal amplitudes show reduced multimode interference and the receptivity coefficients have been computed to have fewer oscillations. With an overall greater magnitude, the receptivity coefficients suggest a possible earlier transition location than the previous numerical study by He & Zhong (2023 J. Spacecr. Rockets, vol. 60, no. 6, pp. 1927–1938). Additionally, a discrete entropy-layer mode is recovered, contributing to instability development alongside modes F and S. These findings support the use of bi-orthogonal decomposition as a practical tool for receptivity analysis and enhancement of the amplitude method in transition prediction.

The nonlinear growth of perturbations in hydrodynamic interfacial instabilities can be of particular importance in both scientific research (e.g. supernova explosion) and engineering applications (e.g. inertial confinement fusion). One of the most significant issues in these instabilities is the long-time nonlinear bubble evolution of a single-mode Rayleigh–Taylor instability (RTI), which remains as an unsolved and challenging problem since Taylor’s seminal work more than seven decades ago. Introduced in this paper is an analytical model for the long-time evolution of bubble velocity, curvature and vorticity, which is established by considering the vorticity accumulation around the bubble in a bilaterally rotational flow system under the classical planar potential-flow theory framework. The proposed theoretical model incorporates not only the classical linear, nonlinear and quasi-steady stages, but the late re-acceleration stage. Meanwhile, the new model can capture the phenomenon of secondary velocity saturation following the stage of bubble re-acceleration. The good agreement between the present model and numerical simulations for all density ratios and dimensions confirms that the accumulation in vorticity tends to break the early stage buoyancy-drag equilibrium mechanism and leads to the establishment of a new equilibrium in the late-stage RTI.

Asymmetries and anisotropies are widespread in biological systems, including in the structure and dynamics of cilia and eukaryotic flagella. These microscopic, hair-like appendages exhibit asymmetric beating patterns that break time-reversal symmetry needed to facilitate fluid transport at the cellular level. The intrinsic anisotropies in ciliary structure can promote preferential beating directions, further influencing their dynamics. In this study, we employ numerical simulation and bifurcation analysis of a mathematical model of a filament driven by a follower force at its tip to explore how intrinsic curvature and direction-dependent bending stiffnesses impact filament dynamics. Our results show that while intrinsic curvature is indeed able to induce asymmetric beating patterns when filament motion is restricted to a plane, this beating is unstable to out-of-plane perturbations. Furthermore, we find that a three-dimensional whirling state seen for isotropic filament dynamics can be suppressed when sufficient asymmetry or anisotropy are introduced. Finally, for bending stiffness ratios as low as 2, we demonstrate that combining structural anisotropy with intrinsic curvature can stabilise asymmetric beating patterns, highlighting the crucial role of anisotropy in ciliary dynamics.

The motionless conducting state of liquid-metal convection with an applied vertical magnetic field confined in a vessel with insulating sidewalls becomes linearly unstable to wall modes through a supercritical pitchfork bifurcation. Nevertheless, we show that the transition proceeds subcritically, with stable finite-amplitude solutions with different symmetries existing at parameter values beneath this linear stability threshold. Under increased thermal driving, the branch born from the linear instability becomes unstable and solutions are attracted to the most subcritical branch, which follows a quasiperiodic route to chaos. Thus, we show that the transition to turbulence is controlled by this subcritical branch and hence turbulent solutions have no connection to the initial linear instability. This is further quantified by observing that the subcritical equilibrium solution sets the spatial symmetry of the turbulent mean flow and thus organises large-scale structures in the turbulent regime.

Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich’s $f$-plane extension of the Miyata–Maltseva–Choi–Camassa model. We develop simultaneous asymptotic expansions for the evolving mean fields and deviations from them to describe a large class of uni-directional waves via the Ostrovsky equation, which fully decouples from mean-field variations. The latter generate additive inertial oscillations in the shear and in the phase of both the interfacial displacement and shear. Unlike conventional derivations leading to the Ostrovsky equation, our formulation does not impose the zero-mean constraints on the initial conditions of any variable. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg–de Vries (KdV) equation but with local defects, both with and without rotation. We show that rotation leads to the emergence of bursts of internal waves and shear currents, qualitatively similar to the wavepackets generated from solitons and modulated cnoidal waves in earlier studies, but emerging much faster. We also show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the KdV equation (appropriately understood), as well as the Weirstrass–Erdmann corner condition for broken extremals of the associated variational problem and a natural weak formulation. Being smoothed in numerical simulations, they behave, in the absence of rotation, as long-lived states with no visible evolution, while rotation leads to the emergence of strong bursts.

We study the stability of a steady Eckart streaming jet flowing in a closed cylindrical cavity. This configuration is a generic representation of industrial processes where driving flows in a cavity by means of acoustic forcing offers a contactless way of stirring or controlling flows. Successfully doing so, however, requires sufficient insight into the topology induced by the acoustic beam. This, in turn, raises the more fundamental question of whether the basic jet topology is stable and, when it is not, of the alternative states that end up being acoustically forced. To answer these questions, we consider a flow forced by an axisymmetric diffracting beam of attenuated sound waves emitted by a plane circular transducer at one cavity end. At the opposite end, the jet impingement drives recirculating structures spanning nearly the entire cavity radius. We rely on linear stability analysis (LSA) together with three-dimensional nonlinear simulations to identify the flow destabilisation mechanisms and to determine the bifurcation criticalities. We show that flow destabilisation is closely related to the impingement-driven recirculating structures, and that the ratio $C_R$ between the cavity and the maximum beam radii plays a key role on the flow stability. In total, we identified four mode types destabilising the flow. For $4 \leqslant C_R \leqslant 6$, a non-oscillatory perturbation rooted in the jet impingement triggers a supercritical bifurcation. For $C_R = 3$, the flow destabilises through a subcritical non-oscillatory bifurcation and we explain the topological change of the unstable perturbation by analysing its critical points. Further reducing $C_R$ increases the shear within the flow and gradually moves the instability origin to the shear layer between impingement-induced vortices: for $C_R = 2$, an unstable travelling wave grows out of a subcritical bifurcation, which becomes supercritical for $C_R=1$. For each geometry, the nonlinear three-dimensional (3-D) simulations confirm both the topology and the growth rate of the unstable perturbation returned by LSA. This study offers fundamental insight into the stability of acoustically driven flows in general, but also opens possible pathways to either induce turbulence acoustically or to avoid it in realistic configurations.

We use scanning-tomographic particle image velocimetry introduced by Casey, Sakakibara & Thoroddsen (Phys. Fluids, vol. 25 (2), 2013, p. 025102) to measure the volumetric velocity field in a fully turbulent round jet. The experiments are performed for ${Re}=2640,\, 5280$ and $10\,700.$ Using Fourier-based proper orthogonal decomposition (POD), the dominant modes that describe the velocity and vorticity fields are extracted. We employ a new method of averaging POD modes from different experimental runs using a phase-synchronisation with respect to a common basis. For the dominant azimuthal wavenumber $m=1,$ the first and second POD modes of the axial velocity have opposite signs and appear as embracing helical structures, with opposite twist, while, for the same parameters, POD modes of the radial velocity extend to the axis of symmetry and, interestingly, also show a helical shape. The $(m=1)$-POD modes for the azimuthal vorticity appear as two separate structures, consisting of C-shaped loops in the region away from the axis and helically twisted axial tubes close to the axis. The corresponding axial vorticity modes are cone-like and appear as inclined streaks of alternate sign in the $r$–$z$-plane, similar to velocity streaks seen in wall-bounded shear flows. Temporal analysis of the dynamics shows that a $(m=1)$ two-mode velocity POD representation precesses with a Strouhal number of approximately $St=0.05,$ while the same reconstruction based on vorticity POD modes has a slightly higher Strouhal number of $St=0.06.$

The inviscid mechanism, driving flow instabilities in a $1:3$, planar and symmetric sudden expansion, is discerned through a sensitivity-based protocol, also referred to as inviscid structural sensitivity analysis, with a specific focus on the onset and nature of the secondary instability. The fundamental idea of this methodology is to change the contribution of viscosity solely in the global stability equations, while freezing the base-flow field at the critical conditions. This is practically implemented by decoupling the Reynolds number that serves as the control parameter for determining the steady base flow from that governing the disturbance evolution, in order to repeat the structural sensitivity analysis while progressively increasing the Reynolds number in the linearised equations only. Accordingly, the sequence of structural sensitivity maps enables us to highlight the flow regions where the inviscid instability mechanism acts. The numerical results reveal that the classical structural sensitivity analysis accurately locates the wavemaker region within the primary recirculation zone, but only its inviscid limit can unveil that the core of the instability coincides with the centre of the primary vortex: a hallmark of an elliptic instability. To validate the global findings, the results of the inviscid structural sensitivity analysis are compared with those obtained from geometric optics. The agreement of the two approaches confirms the inviscid character of the instability, thereby providing a complete picture of the nature of the secondary bifurcation.

The forced breakup of liquid jets in ambient gas surroundings is studied systematically through numerical simulations and theoretical analyses, with particular emphasis on characterising the response modes of jet breakup across wide ranges of perturbation frequency and amplitude. Simulations reveal that the breakup of liquid jet can be effectively synchronised with external actuation within a frequency range encompassing the natural breakup frequency, thereby enabling the generation of highly uniform droplets. As the perturbation frequency exceeds an upper critical value, the external perturbation cannot dominate the jet breakup, while below a lower critical frequency, the jet breaks up with multiple droplets generated within one period. A high perturbation amplitude can result in liquid accumulation, leading to the formation of a pancake-shaped jet configuration. Through spectrum analyses, the development of jet interface perturbations under different response modes is elucidated, revealing the competition between the natural frequency and the external frequency. A linear instability analysis of a liquid jet is performed, which successfully predicts the synchronised frequency range by comparing the breakup time between the free liquid jet and the actuated jet, along with the variation tendencies of jet breakup length with varying perturbation frequency, amplitude and jet velocity. Quantitative numerical results demonstrate that in the case of multiple droplet generation under low perturbation frequency, the rear droplet maintains a higher velocity than its leading counterpart and the emergence of a high-pressure zone at the leading edge of a droplet train facilitates the droplet coalescence. Furthermore, the study introduces an innovative approach by superimposing periodic pulses onto the sinusoidal perturbation waveform, enabling active modulation of multiple droplet merging dynamics. This fundamental study is intended to offer valuable guidance for the on-demand generation of droplets in various industrial applications.